Publication: Heegaard splittings of prime 3-manifolds are not unique
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1976
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Michigan Mathematical Journal
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The authors construct an infinite family of prime homology 3-spheres of Heegaard genus 2, satisfying the following two non-uniqueness properties: (1) Each of the manifolds can be structured as the 2-fold cyclic branched cover over each of two inequivalent knots, one of which is a torus knot. (2) Each of the manifolds admits at least two equivalence classes of genus 2 Heegaard splittings. All of the manifolds are Seifert fiber spaces, the properties of which are used to prove (1). The non-uniqueness of Heegaard splittings is based on the work of the first author and H. M. Hilden [Trans. Amer. Math. Soc. 213 (1975), 315–352], who proved that for Heegaard genus 2 splittings of the 2-fold branched cyclic cover of the knot K, the equivalence class of the Heegaard splitting determines uniquely the knot type K. The authors then show that if Σp,q is the 2-fold cyclic branched cover of the torus knot (p,q), then Σp,q is also the 2-fold cyclic branched cover of a knot different from (p,q), and that Σp,q admits a Heegaard splitting of genus 2.
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J. S. Birman, On the equivalence of Heegaard splittings of closed, orientable 3-manifolds. In Knots, groups, and 3-manifolds (L. P. Neuwirth,Editor).Princeton Univ. Press, Annals of Math. Studies No. 84 (1975), 137-164.
J. S. Birman and H. M. Hilden, The homeomorphism problem for S3. Bull.Amer. Math. Soco 79 (1973), 1006-1010.
J. S. Birman and H. M. Hilden, Heegaard splittings 01 branched coverings 01 S3. Trans. Amer. Math. Soc. 213 (1975), 315-352.
R. Engmann, Nicht-homoomorphe Heegaard-Zerlegungen vom Geschlecht 2 der zusammenhangenden Summe zweier Linsenraume. Abh. Math. Sem. Univ. Hamburg 35 (1970), 33-38.
R. H. Fox, A quick trip through knot theory. In Topology of 3-manifolds and related topies (Prac. The Univ. of Georgia Institute, 1961), pp. 120-167. (M.K.Fort, Jr., Editor) Prentice-Hall, Englewood Cliffs, N.J., 1962.
R. E. Goodrick, Numerieal invariantsof of knots. Illinois J. Math. 14 (1970), 414-418.
J. M. Montesinos, Sobre la conjeetura de Poinearé y los reeubridores ramificados sobre un nodo. Tesis, Facultad de Ciencias, Universidad Complutense de Madrid, 1972.
J. M. Montesinos, Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas. Bol. Soc. Mat. Mexicana (2) 18 (1973), 1-32.
L. Neuwirth, The algebraic determination of the genus of knots. Amer. J. Math. 82 (1960), 791-798.
K. Reidemeister, Zur dreidimensionalen Topologie. Abh. Math. Serna Univ. Hamburg 9 (1933), 189-194.
H. Schubert, Über eine numerisehe Knoteninvariante. Math., Z. 61 (1954), 245-288.
H. Seifert, Topologie dreidimensionaler gelaserter Raume. Acta Math. 60 (1933), 147-238.
H. Seifert, Über das Geschlecht von Knoten. Math. Ann. 110 (1934), 571-592.
J. Singer, Three-dimensional manifolds and their Heegaard diagrams. Trans. Amer. Math. Soc. 35 (1933), 88-111.
O. Ja. Viro, Linkings, 2-sheeted branched coverings, and braids. Mat. Sb. (N.S.) 87 (129) (1972), 216-228. English translation: Math. USSR-Sb. 16 (1972), 223-236.
F. Waldhausen, Eine Klasse von 3-dimensionalen Mannigfaltigkeiten. II. Invent. Math. 4 (1967), 87-117.
F. Waldhausen, Heegaard-Zerlegungen der 3-Sphare. Topology 7 (1968), 195-203.
F. Waldhausen , Über lnvolutionen der 3-Sphiire. Topology 8 (1969), 81-91.